In this paper, we investigate an reconfigurable intelligent surface (RIS)-aided integrated sensing and communication (ISAC) system. Our objective is to maximize the achievable sum rate of the multi-antenna communication users through the joint active and passive beamforming. Weighted minimum mean-square error (WMMSE) method is used to reformulate the original problem into an equivalent one. Then, we utilize an alternating optimization (AO) approach to separate the optimization variables and decompose this challenging problem into two subproblems. Given reflecting coefficients, a penalty-based algorithm is utilized to deal with the transmit power and the non-convex radar signal-to-noise ratio (SNR) constraints. For the given beamforming matrix of the BS, we apply majorization-minimization (MM) to transform the problem into a quadratic constraint quadratic programming (QCQP) problem, which is ultimately solved using a semidefinite relaxation (SDR)-based algorithm. Simulation results illustrate the advantage of deploying RIS in the considered multi-user MIMO (MU-MIMO) ISAC systems.